Assumption REF  of Corollary REF  implies that the process under consideration is non-stationary and we subsequently find that we can recover the root. We further show in example REF  that in the stationary, non-reversible case it is not possible to recover the root. This seems to fit with the collective empirical evidence of [1]}, [2]}, and [3]}. That is, in some cases a non-stationary process can recover the root, whereas a stationary, non-reversible process cannot. Assumption REF  is also the source of a subtle and interesting contradiction. It can be shown under mild assumptions that the product of arbitrary stochastic matrices will eventually have almost identical rows as the number of terms in the product increases [4]}. It is not difficult to show that the rows must be the stationary distribution of the product, and that if the terms are strand symmetric then the product must be strand symmetric. So for any reasonably long history of strand-symmetric processes, we must have that the marginal probabilities are also strand-symmetric. The conclusion is that for our model to work, the process should probably have been strand asymmetric prior to the root of the tree, and strand symmetric thereafter. An alternative interpretation is that our model is just a model, and that we could start by assuming that the root probabilities are sufficiently asymmetric for the process to be non-stationary, and then that the process is sufficiently strand-symmetric for the model to be approximately correct. In either case assumption REF  of Corollary REF  could be an interesting tool for probing the limits of inference of this model.
